cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A273977 Words over an alphabet of size 9 that are in standard order with at least one letter repeated.

Original entry on oeis.org

11, 111, 112, 121, 122, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11233, 11234, 12111, 12112, 12113, 12121, 12122, 12123, 12131, 12132, 12133, 12134, 12211, 12212
Offset: 1

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Author

Daniel Devatman Hromada, Nov 10 2016

Keywords

Comments

We study words made of letters from an alphabet of size b, where b >= 1. (Here b=9.) We assume the letters are labeled {1,2,3,...,b}. There are b^n possible words of length n.
We say that a word is in "standard order" if it has the property that whenever a letter i appears, the letter i-1 has already appeared in the word. This implies that all words begin with the letter 1.
These are the words described in row b=9 of the array in A278986.
This sequence can be potentially expanded by a much more efficient algorithm than the brute-force one presented in the program section.

References

  • Daniel Devatman Hromada, Integer-based nomenclature for the ecosystem of repetitive expressions in complete works of William Shakespeare, submitted to special issue of Argument and Computation on Rhetorical Figures in Computational Argument Studies, 2016.

Links

Crossrefs

Cf. A278987.

Programs

  • Mathematica
    Select[Range[2*10^4], And[Max[DigitCount@ #] >= 2, Range@ Length@ Union@ # == DeleteDuplicates@ # &@ IntegerDigits@ #] &] (* Michael De Vlieger, Nov 10 2016 *)

Extensions

Edited by N. J. A. Sloane, Dec 06 2016
Duplicated terms removed from b-file by Andrew Howroyd, Feb 27 2018

A273978 List of words of length n over an alphabet of size 9 that are in standard order and which have the property that every letter that appears in the word is repeated.

Original entry on oeis.org

11, 111, 1111, 1122, 1212, 1221, 11111, 11122, 11212, 11221, 11222, 12112, 12121, 12122, 12211, 12212, 12221, 111111, 111122, 111212, 111221, 111222, 112112
Offset: 1

Views

Author

Daniel Devatman Hromada, Nov 10 2016

Keywords

Comments

We study words made of letters from an alphabet of size b, where b >= 1. (Here b=9.) We assume the letters are labeled {1,2,3,...,b}. There are b^n possible words of length n.
We say that a word is in "standard order" if it has the property that whenever a letter i appears, the letter i-1 has already appeared in the word. This implies that all words begin with the letter 1.
These are the words described in row b=9 of the array in A278987.

References

  • D. D. Hromada, Integer-based nomenclature for the ecosystem of repetitive expressions in complete works of William Shakespeare, submitted to special issue of Argument and Computation on Rhetorical Figures in Computational Argument Studies, 2016.

Links

Crossrefs

Subset of A273977.
Cf. A278987.

Extensions

Edited by N. J. A. Sloane, Dec 06 2016

A278988 a(n) is the number of words of length n over an alphabet of size 3 that are in standard order and which have the property that every letter that appears in the word is repeated.

Original entry on oeis.org

0, 0, 1, 1, 4, 11, 41, 162, 610, 2165, 7327, 23948, 76352, 239175, 739909, 2268710, 6912430, 20966441, 63390587, 191220048, 575888044, 1732382363, 5207108161, 15642295562, 46970926394, 141005053341, 423208097431, 1270026944852, 3810919694680, 11434503913775, 34307135619197
Offset: 0

Views

Author

N. J. A. Sloane, Dec 06 2016

Keywords

Links

Crossrefs

A row of the array in A278987.

Formula

Conjectures from Colin Barker, Nov 25 2017: (Start)
G.f.: x^2*(1 - 9*x + 34*x^2 - 71*x^3 + 100*x^4 - 97*x^5 + 52*x^6 - 12*x^7) / ((1 - x)^3*(1 - 2*x)^2*(1 - 3*x)).
a(n) = (2*(3+3^n) - 3*(2+2^n)*n + 6*n^2) / 12 for n>3.
a(n) = 10*a(n-1) - 40*a(n-2) + 82*a(n-3) - 91*a(n-4) + 52*a(n-5) - 12*a(n-6) for n>9.
(End)

A278989 a(n) is the number of words of length n over an alphabet of size 4 that are in standard order and which have the property that every letter that appears in the word is repeated.

Original entry on oeis.org

0, 0, 1, 1, 4, 11, 41, 162, 715, 3425, 16777, 80928, 379347, 1726375, 7654817, 33219630, 141692075, 596122477, 2480969257, 10237751324, 41963944275, 171103765747, 694775280993, 2812004330666, 11352134320523, 45736973060601, 183981143571721, 739167464021912, 2966826380664595, 11899055223201855
Offset: 0

Views

Author

N. J. A. Sloane, Dec 06 2016

Keywords

Links

Crossrefs

A row of the array in A278987.

Formula

Conjectures from Colin Barker, Nov 25 2017: (Start)
G.f.: x^2*(1 - 19*x + 159*x^2 - 776*x^3 + 2474*x^4 - 5498*x^5 + 8993*x^6 - 11471*x^7 + 11815*x^8 - 9478*x^9 + 5348*x^10 - 1848*x^11 + 288*x^12) / ((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2*(1 - 4*x)).
a(n) = 20*a(n-1) - 175*a(n-2) + 882*a(n-3) - 2835*a(n-4) + 6072*a(n-5) - 8777*a(n-6) + 8458*a(n-7) - 5204*a(n-8) + 1848*a(n-9) - 288*a(n-10) for n > 14.
(End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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